the tube length of the concentric tube heat exchanger is approximately 25.50 meters.
the tube length (L) of the counterflow, concentric tube heat exchanger step by step:
Given information:
- Inner tube diameter (di) = 20 mm = 0.02 meters
- Outer annulus diameter (do) = 40 mm = 0.04 meters
- Flow rate of water (H2O) through the inner tube (m_dot_water) = 0.18 kg/s
- Flow rate of engine oil through the outer annulus (m_dot_oil) = 0.12 kg/s
- Inlet temperature of oil (T_inlet_oil) = 95°C = 95 + 273.15 K = 368.15 K
- Outlet temperature of oil (T_outlet_oil) = 65°C = 65 + 273.15 K = 338.15 K
- Inlet temperature of water (H2O) (T_inlet_water) = 30°C = 30 + 273.15 K = 303.15 K
- Neglecting tube wall resistance, fouling factors, and heat loss to surroundings.
Step 1: Calculate the heat transfer rate (Q):
We'll start by calculating the heat transfer rate (Q) between the oil and water using the formula:
Q = m_dot_oil * Cp_oil * (T_inlet_oil - T_outlet_oil)
Where:
- m_dot_oil = Mass flow rate of oil = 0.12 kg/s
- Cp_oil = Specific heat capacity of oil (assumed constant) = 2,000 J/(kg·K)
Q = 0.12 kg/s * 2,000 J/(kg·K) * (368.15 K - 338.15 K)
Q = 0.12 kg/s * 2,000 J/(kg·K) * 30 K
Q = 72,000 J/s (or 72 kW)
Step 2: Calculate the logarithmic mean temperature difference (ΔT_lm):
In a counterflow heat exchanger, ΔT_lm can be calculated as:
ΔT_lm = (ΔT1 - ΔT2) / ln(ΔT1 / ΔT2)
Where:
- ΔT1 = T_inlet_oil - T_inlet_water
- ΔT2 = T_outlet_oil - T_outlet_water
Assuming the water temperature remains constant at 30°C:
ΔT1 = (368.15 K - 303.15 K)
ΔT2 = (338.15 K - 303.15 K)
Now, calculate ΔT_lm:
ΔT_lm = [(ΔT1 - ΔT2) / ln(ΔT1 / ΔT2)]
ΔT_lm = [(368.15 K - 303.15 K - (338.15 K - 303.15 K)) / ln((368.15 K - 303.15 K) / (338.15 K - 303.15 K))]
Step 3: Calculate the heat transfer surface area (A):
Assume a typical overall heat transfer coefficient (U). The value of U depends on the specific design and material properties. For demonstration purposes, let's assume U = 500 W/(m^2·K).
A = Q / (U * ΔT_lm)
A = 72,000 W / (500 W/(m^2·K) * ΔT_lm)
Step 4: Calculate the tube length (L):
To find L, we need to know the length of the inner tube over which the heat transfer takes place. Assuming the entire length of the inner tube is used for heat transfer:
L = A / (π * di)
L = (72,000 W) / (500 W/(m^2·K) * ΔT_lm * π * 0.02 m)
Now, calculate L using the values provided. This will give you the tube length of the concentric tube heat exchanger.
Let's calculate the tube length (L) step by step:
Step 1: Calculate the heat transfer rate (Q):
Q = m_dot_oil * Cp_oil * (T_inlet_oil - T_outlet_oil)
Q = 0.12 kg/s * 2,000 J/(kg·K) * (368.15 K - 338.15 K)
Q = 0.12 kg/s * 2,000 J/(kg·K) * 30 K
Q = 72,000 J/s (or 72 kW)
Step 2: Calculate the logarithmic mean temperature difference (ΔT_lm):
ΔT1 = T_inlet_oil - T_inlet_water
ΔT1 = 368.15 K - 303.15 K
ΔT1 = 65 K
ΔT2 = T_outlet_oil - T_outlet_water
ΔT2 = 338.15 K - 303.15 K
ΔT2 = 35 K
Now, calculate ΔT_lm using the formula:
ΔT_lm = (ΔT1 - ΔT2) / ln(ΔT1 / ΔT2)
ΔT_lm = (65 K - 35 K) / ln(65 K / 35 K)
ΔT_lm ≈ 45.23 K
Step 3: Calculate the heat transfer surface area (A):
Assume U = 500 W/(m^2·K).
A = Q / (U * ΔT_lm)
A = 72,000 W / (500 W/(m^2·K) * 45.23 K)
A ≈ 3.193 m^2
Step 4: Calculate the tube length (L):
L = A / (π * di)
L = 3.193 m^2 / (π * 0.02 m)
L ≈ 25.50 meters
So, the tube length of the concentric tube heat exchanger is approximately 25.50 meters.